this question was repeatd six times in a various exams between 1971 to 2001. if C_0 ,C_1 ,C_2 .......,C_n are the coefficients in the expansion of (1+x)^n then c_0 +2C_1 +3C_2 ........(n+1)C


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Question Number 106690 by M±th+et+s last updated on 06/Aug/20

$t h i s q u e s t i o n w a s r e p e a t d s i x t i m e s$ $i n a v a r i o u s e x a m s b e t w e e n 1971 t o 2001 .$ $i f C_{0}, C_{1}, C_{2} . . . . . . ., C_{n} a r e t h e c o e f f i c i e n t s$ $i n t h e e x p a n s i o n o f {(1 + x)}^{n} t h e n$ $c_{0} + 2 C_{1} + 3 C_{2} . . . . . . . . (n + 1) C_{n} = ?$
	Answered by prakash jain last updated on 06/Aug/20

	$(r + 1) C_{r} = {r C}_{r} + C_{r}$ ${(1 + x)}^{n} = \underset{j = 0}{\sum^{n}} C_{j} x^{j}$ $d i f f e r e n t i a t e$ $n {(1 + x)}^{n - 1} = \underset{j = 0}{\sum^{n}} {j C}_{j} x^{j - 1}$ $x = 1$ $n 2^{n - 1} = \underset{j = 0}{\sum^{n}} {j C}_{j}$ $\underset{r = 0}{\sum^{n}} (r + 1) C_{r} = \underset{r = 0}{\sum^{n}} {r C}_{r} + \underset{r = 0}{\sum^{n}} C_{r}$ $= n 2^{n - 1} + 2^{n}$
	Commented by M±th+et+s last updated on 06/Aug/20

	$w e l l d o n e s i r$
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